Matrix exponential

Definition

$$\begin{eqnarray} e^{At} \triangleq I + At + A^2\frac{t^2}{2!} + ... = \sum_{i=0}^\infty A^i \frac{t^i}{i!} \\ \frac{de^{At}}{dt} = A [I + At + A^2 \frac{t^2}{2!} + ...] = Ae^{At} \end{eqnarray}$$

Properties

• $e^{A0} = I + A0 + A^2 \frac{0^2}{2!} + ... = I$
• $(e^{At})^{-1} = e^{-At}$
• $\frac{d}{dt} e^{At} = A e^{At}$

Notes

If $A$ is $1 \times 1$ i.e. scalar, this becomes the power series, $$e^{at} = \sum_{k=0}^\infty \frac{(at)^k}{k!} = 1 + at + a^2\frac{t^2}{2!} + ...$$ which matches a common definition of the exponential function

References

1. https://crrl.poly.edu/6253/lectures/lect4.pdf
2. https://en.wikipedia.org/wiki/Matrix_exponential
3. https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function